Munich Personal RePEc Archive

A Simplified Approach to Analyzing

Multi-regional Core-Periphery Models

Akamatsu, Takashi and Takayama, Yuki

18 August 2009

Online at https://mpra.ub.uni-muenchen.de/21739/

MPRA Paper No. 21739, posted 31 Mar 2010 05:48 UTC

1

A Simplified Approach to Analyzing Multi-regional

Core-Periphery Models

Takashi Akamatsu*1

and Yuki Takayama*

Abstract This paper shows that the evolutionary process of spatial agglomeration in

multi-regional core-periphery models can be explained analytically by a much

simpler method than the continuous space approach of Krugman (1996). The

proposed method overcomes the limitations of Turing’s approach which has been

applied to continuous space models. In particular, it allows us not only to examine

whether or not agglomeration of mobile factors emerges from a uniform

distribution, but also to trace the evolution of spatial agglomeration patterns (i.e.,

bifurcations from various polycentric patterns as well as from a uniform pattern)

with decreases in transportation cost.

Keywords: agglomeration, core-periphery model, multi-regional, stability, bifurcation

JEL classification: R12, R13, F15, F22, C62

1. Introduction

More than a decade has passed since the new economic geography (NEG) emerged with

now well-known modeling techniques such as “Dixit-Stiglitz, Icebergs, Evolution, and the

Computer”, by Fujita, Krugman and Venables (1999). These new modeling techniques, first

introduced in the core-periphery (CP) model developed by Krugman (1991), provided a

full-fledged general equilibrium approach and led to numerous studies on extending the

original framework. Furthermore, in recent years, there has been a proliferation of theoretical

and empirical work applying the NEG framework to deal with various policy issues (Baldwin

et al., (2003), Behrens and Thisse (2007), Combes et al., (2009)).

Despite the remarkable growth of the NEG theory, there remain some fundamental issues

that need to be addressed before the theory provides a sound foundation for empirical work

and practical applications. One of the most relevant issues is to reintroduce spatial aspects

into the theory. Even though this direction was pursued in the early development stages of

NEG (e.g., Krugman (1993, 1996), Fujita et al., (1999)), recent theoretical studies have been

almost exclusively limited to the two-region CP/NEG models in which many essential aspects

of “space/geography” have almost vanished. As a result, little is known about the rich

properties of the multi-regional CP model. In view of the fact that two-region analysis has

1* Graduate School of Information Sciences, Tohoku University, Sendai, Japan.

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some serious limitations2 due to the “degeneration of space”, it seems reasonable to argue

that “a theoretical analysis of economic geography must make an effort to get beyond the

two-location case” since “real-world geographical issues cannot be easily mapped into

two-regional analysis” (Fujita et al., (1999, Chap.6)). In other words, advancing our

understanding of the multi-regional NEG/CP models is a prerequisite for systematic empirical

work as well as for systematic evaluation of policy proposals.

Why, despite the obvious needs, have there been very few theoretical studies on

multi-regional CP models in the last decade? This seems to be a direct result of technical

difficulties that inevitably arise in examining the properties of the multi-regional CP model.

As is well known in the NEG theory, the two-regional CP model, depending on transportation

costs, exhibits a “bifurcation” from a symmetric equilibrium to an asymmetric equilibrium. In

dealing with the multi-regional CP model, we are likely to encounter more complex

bifurcation phenomena and hence we need to devise better methods to analyze them. In

contrast to the large number of works on the CP model that have flourished during the last

decade, there has been very little progress in developing effective approaches to this

bifurcation problem since the work of Krugman (1996) and Fujita et al., (1999).

The only method that has been used to analyze bifurcation in the multi-regional CP model

is the Turing (1952) approach, in which one focuses on the onset of instability of a uniform

equilibrium distribution (“flat earth equilibrium”) of mobile agents. That is, assuming a

certain class of adjustment process (e.g., “replicator dynamics”), one examines a trend of the

economy away from, rather than toward, the flat earth equilibrium whose instability implies

the emergence of some agglomeration.3 Krugman (1996) and Fujita et al., (1999, Chap.6)

applied this approach to the CP model with a continuum of locations on the circumference

and succeeded in showing that the steady decreases in transportation costs lead to the

instability of the flat earth equilibrium state. Recently, a few studies have also applied this

approach, and re-examined the robustness of the Krugman’s findings in the CP model with

continuous space racetrack economy. Mossay (2003) theoretically qualifies the Krugman’s

results in the case of workers’ heterogeneous preferences for location. More recently, Picard

and Tabuchi (2009) examined the impact of the shape of transport costs on the structure of

spatial equilibria4.

2 For more elaborated discussions on the limitations of the two-regional analysis, see, for example, Fujita

and Thisse (2009), Akamatsu et al., (2009), Behrens and Thisse (2007), Fujita and Krugman (2004). 3 The first notable application of this approach to analyzing agglomeration in a spatial economy was made

by Papageorgiou and Smith (1983).

4 Tabuchi et al. (2005) also study the impact of falling transport costs on the size and number of cities in a

multi-regional model that extends a two-regional CP model by Ottaviano et al., (2002). Oyama (2009)

showed that the multi-regional CP model admits a potential function, which allowed to identify a stationary

state that is uniquely absorbing and globally accessible under the perfect foresight dynamics. However,

these analyses are restricted to a very special class of transport geometry in which regions are pairwise

equidistant.

3

While this approach offers a remarkable way of thinking about a seemingly complex issue,

it has two important limitations. First, it deals with only the first stage of agglomeration when

the value of a parameter (e.g., transportation cost) steadily changes; it cannot give a good

description of what happens thereafter. Indeed, Krugman (1996) and Fujita et al., (1999,

Chap.17) resort to rather ad hoc numerical simulations for analyzing the possible bifurcations

in the later stages; recent studies of Mossay (2003) and Picard and Tabuchi (2009) are silent

on the bifurcations in the later stages. Second, the eigenvalue analysis required in the

approach becomes complicated, and it is, in general, almost impossible to analytically obtain

the eigenvalues for an arbitrary configuration of mobile workers. This is one of the most

difficult obstacles that prevent us from understanding the general properties of the

multi-regional CP model.

In this paper, we show that the evolutionary process of spatial agglomeration in the

multi-regional CP models can be readily explained by a much simpler method than the

continuous space approach of Krugman (1996) and Fujita et al., (1999). The main features of

the proposed method are as follows:

1) it is applicable to the CP model with an arbitrary discrete number of regions, in contrast

to Krugman’s approach that is restricted to a special limiting case (i.e., continuous space).

2) it exploits the concept of a “spatial discounting matrix (SDM)” in a circular city/region

system (“racetrack economy (RE)”). This together with the discrete Fourier transformation

(DFT) provides an analytically tractable method of elucidating the agglomeration properties

of the multi-regional CP model, without resorting to numerical techniques.

3) it allows us not only to examine whether or not agglomeration of mobile factors emerges

from a uniform distribution, but also to trace the evolution of spatial agglomeration patterns

(i.e., bifurcations from various polycentric patterns as well as from a uniform pattern) with

the decreases in transportation cost. That is, it overcomes the limitations of Turing’s approach

that Krugman (1996), Fujita et al., (1999), Mossay (2003), and Picard and Tabuchi (2009)

encountered in their continuous space models.

To demonstrate the proposed method, we employed a pair of multi-regional CP models,

which are “solvable” variants of Krugman’s original CP model. By the term “solvable”, we

mean that an explicit form of the indirect utility function of a consumer for a short-run

equilibrium (in which a location pattern of workers is fixed) can be obtained (Proposition 1).

More specifically, each of the CP models presented here is a multi-regional version of the

two-region CP models recently developed by Forslid and Ottaviano (2003) and Pflüger

(2004).

In the analysis of these CP models, we intentionally restricted ourselves to the case of four

regions for clarity of exposition, although the approach presented in this paper can deal with a

model with an arbitrary number of regions. Interested readers can consult Akamatsu et al.,

(2009) for more general cases. The four-region setting allowed us to illustrate the essential

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feature of our approach without going into too much technical detail. Indeed, even in this

simple setting, we observed a number of interesting properties of the multi-regional CP model

that are not reported in the literature.

In order to understand the bifurcation mechanism of the CP model, we need to know how

the eigenvalues of the Jacobian matrix of the adjustment process depend on bifurcation

parameters (e.g., the transportation cost coefficient τ). A combination of the RE (with discrete

locations) and the resultant circulant properties of the SDM greatly facilitate this analysis.

Indeed, it is shown (in Proposition 2) that the eigenvalue gk of the Jacobian matrix of the

adjustment process can be expressed as a quadratic function of the eigenvalue fk of the SDM.

The former eigenvalue gk thus obtained has a natural economic interpretation as the strength

of “net agglomeration force”, and offers the key to understanding the agglomeration

properties of the CP economy.

To investigate the evolutionary process of the spatial agglomeration in the multi-regional

CP models, we considered the process in which the value of the spatial discounting factor

(SDF) r steadily increased (which means transportation cost decreases) over time. Starting

from r = 0 at which a uniform distribution of skilled labor is a stable equilibrium state, we

investigated when and what spatial patterns of agglomeration emerged (i.e. a bifurcation

occuring) with the increases in the SDF. The analytical expression of the eigenvalues allowed

us to identify the “break point” and the associated patterns of agglomeration that emerged at

the bifurcation (Proposition 4). Unlike the conventional two-region models that exhibit only

a single time of bifurcation, this is not the end of the story in the four-region model. Indeed, it

is shown (in Proposition 5) that the agglomeration pattern after the first bifurcation evolves

over time with the steady increases in the SDF: it first grows to a duocentric pattern, which

continues to be stable for a while; further increases in the SDF, however, trigger the

occurrence of a second bifurcation, which in turn leads to the formation of a monocentric

agglomeration. This result was derived by using a simple analytical technique based on a

similarity transformation. Furthermore, it was theoretically deduced (in Proposition 6) that

the collapse of agglomeration (that corresponds to “re-dispersion” in the two-region CP

model) can occur for a high-SDF range.

The remainder of the paper is organized as follows. Section 2 presents the equilibrium

conditions of the multi-regional CP models as well as definitions of the stability and

bifurcation of the equilibrium state. Section 3 defines the SDM in a racetrack economy, whose

eigenvalues are provided by a DFT. Section 4 analyzes the evolutionary process of spatial

patterns observed in our models. Section 5 concludes the paper.

2. The Model

2.1. Basic Assumptions

We present a pair of multi-regional CP models whose frameworks follow Forslid and

5

Ottaviano(2003) and Pflüger(2004) (defined as FO and Pf). The basic assumptions of the

multi-regional CP model are the same as those of the FO and Pf models except for the number

of regions, but we provide them here for completeness. The economy is composed of K

regions indexed by 1,...,1,0 −= Ki , two factors of production and two sectors. The two

factors of production are skilled and unskilled labor. Each worker supplies one unit of his type

of labor inelastically. The skilled worker is mobile across regions and hi denotes the number

of these factors located in region i. The total endowment of skilled workers is H. The

unskilled worker is immobile and equally distributed across all regions. The unit of unskilled

worker is chosen such that the world endowment KL = (i.e., the number of unskilled

workers in each region is one). The two sectors are agriculture (abbreviated by A) and

manufacturing (abbreviated by M ). The A-sector output is homogeneous and produced using

a unit input requirement of unskilled labor under constant returns to scale and perfect

competition. This output is the numéraire and assumed to be produced in all regions. The

M-sector output is a horizontally differentiated product and produced using both skilled and

unskilled labor under increasing returns to scale and Dixit-Stiglitz monopolistic competition.

The goods of both sectors are transported, but transportation of the A-sector goods is

frictionless while transportation of the M-sector goods is inhibited by iceberg transportation

costs. That is, for each unit of the M-sector goods transported from region i to j, only a

fraction 1/1 <ijφ arrives.

All workers have identical preferences U over the M and A-sector goods. The utility of

each consumer in region i is given by:

[FO model5] A

iMi

Ai

Mi CCCCU ln)1(ln),( μμ −+= )10( << μ (2.1)

[Pf model] Ai

Mi

Ai

Mi CCCCU += ln),( μ )0( >μ (2.2)

∑ ∫−

∈− ⎟

⎠⎞

⎜⎝⎛≡

jnk ji

Mi

j

dkkqC

)1/(

/)1()(

σσσσ )1( >σ

where AiC is the consumption of the A-sector goods in region i; M

iC represents the

manufacturing aggregate in region i; )(kq ji is the consumption of variety ],0[ jnk ∈

produced in region j and nj is the number of varieties produced in region j; μ is the constant

expenditure share on industrial varieties and σ is the constant elasticity of substitution

between any two varieties. The budget constraint is given by:

ij

nk jijiAi YdkkqkpC

j

=+ ∑∫ ∈ )()(

where )(kp ji denotes the price in region i of the M-sector goods produced in region j, and Yi

denotes the income of a consumer in region i.

The utility maximization of (2.1) or (2.2) yields the following demand )(kqij of a

5 We take logarithms of the Forslid and Ottaviano(2003) type (i.e., Cobb-Douglas-type) utility function to

facilitate the analysis. Note that this transformation has no influence on the properties of the model.

6

consumer in region i for a variety of the M-sector goods k produced in location j:

[FO model] i

i

ji

ji Ykp

kq σ

σ

ρμ

−

−

=1

)}({)(

[Pf model] σ

σ

ρμ

−

−

=1

)}({)(

i

ji

ji

kpkq

where

∑ ∫−

∈− ⎟

⎠⎞

⎜⎝⎛=

jnk jii

j

dkkp

)1/(1

1)(

σσρ (2.3)

denotes the price index of the differentiated product in region i. Since the total income and

population in region i are wi hi +1 and hi +1, respectively, we have the total demand )(kQ ji :

[FO model] )1()}({

)(1

+= −

−

ii

i

ji

ji hwkp

kQ σ

σ

ρμ

(2.4a)

[Pf model] )1()}({

)(1

+= −

−

i

i

ji

ji hkp

kQ σ

σ

ρμ

(2.4b)

The A-sector technology requires one unit of unskilled labor in order to produce one unit of

output. With free trade in the A-sector, the choice of this goods as the numéraire implies that

in equilibrium the wage of the unskilled worker Liw is equal to one in all regions, that is,

iwLi ∀= 1 . In the M-sector, product differentiation ensures a one-to-one relation between

firms and varieties. Specifically, in order to produce )(kxi unit of product k, a firm incurs a

fixed input requirement of α unit of skilled labor and a marginal input requirement of )(kxiβ

unit of unskilled labor. With 1=Liw , the total cost of production of a firm in region i is thus

given by )(kxw ii βα + , where wi is the wage of the skilled worker. Given the fixed input

requirement α, the skilled labor market clearing implies that in equilibrium the number of

firms is determined by α/ii hn = so that the number of active firms in a region is

proportional to the number of its skilled workers.

Due to the iceberg transportation costs, the total supply of the M-sector firm located in

region i (i.e. )(kxi ) is given by:

∑=j

ijiji kQkx )()( φ (2.5)

Therefore, a typical M-sector firm located in region i maximizes profit as given by:

∑ ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=Π

j jijijiijiji kQwkQkpk )()()()( φβα .

Since we have a continuum of firms, each one is negligible in the sense that its action has no

impact on the market (i.e., the price indices). Hence, the first order condition for the profit

maximization gives:

7

ijij kp φσ

βσ1

)(

−= (2.6)

This expression implies that the price of the M-sector goods does not depend on variety k, so

that )(kQij and )(kxi also do not depend on k. Thus we describe these variables without

argument k. Substituting (2.6) into (2.3), the price index becomes

)1/(1

1

σ

σσβρ

−

⎟⎟⎠

⎞⎜⎜⎝

⎛

−= ∑

jjiji dh (2.7)

where σφ −≡ 1jijid is a “spatial discounting factor” between region i and j: from (2.4), (2.6)

and (2.7), jid is represented as )/()( iiiijiji QpQp , which means that jid is the ratio of total

expenditure in region i for each M-sector product produced in region j to their expenditure for

a domestic product.

2.2. Short-Run Equilibrium

In the short run, the skilled workers are immobile between regions, that is, their spatial

distribution ( T110 ],...,,[ −≡ Khhhh ) is taken as given. The short-run equilibrium conditions

consist of the M-sector goods market clearing condition and the zero profit condition due to

the free entry and exit of firms. The former condition can be written as (2.5). The latter

condition requires that the operating profit of a firm is entirely absorbed by the wage bill of its

skilled workers:

⎟⎟⎠

⎞⎜⎜⎝

⎛−= ∑ )()(

1)( hhh i

jijiji xQpw β

α (2.8)

Substituting (2.4), (2.5), (2.6) and (2.7) into (2.8), we have the short-run equilibrium wage

equations:

[FO model] ∑ +⎟⎟⎠

⎞⎜⎜⎝

⎛=

jjj

j

ij

i hwΔ

dw )1)((

)()( h

hh

σμ

(2.9)

[Pf model] ∑ +⎟⎟⎠

⎞⎜⎜⎝

⎛=

jj

j

ij

i hΔ

dw )1(

)()(

hh

σμ

(2.10)

where kk kjj hdΔ ∑≡)(h denotes the market size of the M-sector in region j. Thus,

)(/ hjij Δd defines the market share in region j of each M-sector product produced in region i.

To obtain the indirect utility function )(hiv , we express the equilibrium wage )(hiw as

an explicit function of h. For this, we rewrite (2.9) and (2.10) in matrix form by using the

“spatial discounting matrix” D whose ),( ji entry is ijd . Then, the equilibrium wage T

110 )](),...,(),([)( hhhhw −≡ Kwww , is given by:

[FO model] )()( )(1

hwMHIhwL

−

⎥⎦⎤

⎢⎣⎡ −=

σμ

σμ

(2.11)

8

[Pf model] )}()({ )( )()(hwhwhw

LH +=σμ

1MhwhMhw )(,)( )()( ≡≡ LH (2.12)

where T]1...,,1,1[≡1 and I is a unit matrix. M and H are defined as

]diag[],diag[,1hHDhΔDΔM ≡≡≡ − (2.13)

This leads to the following proposition.

Proposition 1: The indirect utility T

110 )](),...,(),([)( hhhhv −≡ Kvvv of the multi-regional FO

and Pf models can be expressed as an explicit function of h:

[FO model] )]([ln)()( hwhShv += μ (2.14)

[Pf model] )}()({)()( )()(

1hwhwhShv

LH ++= −σ (2.15)

where T110 ]ln...,,ln,[ln]ln[ −≡ Kwwww . )(),( )(

hwhwH and )()(

hwL are defined in (2.11),

(2.12), and ]ln[)1()( 1DhhS

−−≡ σ .

2.3. Long-Run Equilibrium and Adjustment Dynamics

In the long run, the skilled workers are inter-regionally mobile and will move to the region

where their indirect utility is higher. We assume that they are heterogeneous in their

preferences for location choice. That is, the indirect utility for an individual s located in region

i is expressed as:

)()( )()( sii

si vv ε+= hh

where )(siε denotes the utility representing the idiosyncratic taste for a residential location.

The distribution of }|{ )(

ss

i ∀ε is assumed to be the Weibull distribution and to be identical

and independent across regions. Under this assumption, the fraction Pi(h) of the skilled

workers choosing region i is given by:

∑

≡j j

ii

v

vP

)](exp[

)](exp[)(

h

hh

θθ

(2.16)

where ),0( ∞∈θ is the parameter expressing the inverse of the variance of individual tastes.

When ∞→θ , (2.16) means that the workers decide their location only by )(hiv , which

corresponds to the case without heterogeneity (i.e., the skilled workers are homogeneous).

The long-run equilibrium is defined as the spatial distribution of the mobile workers h that

satisfies the following condition:

iHPh ii ∀= )(h (2.17)

or equivalently written as 0hhPhF =−≡ )()( H , where H is the total endowment of the

skilled worker, and T110 )](...,),(),([)( hhhhP −≡ KPPP . This condition means that the actual

number of individuals hi in each region is equal to the number )(hiHP of individuals who

choose that region under the current distribution h of skilled workers.

9

For this equilibrium condition, it is natural to assume the following adjustment process:

)(hFh =& (2.18)

This is the well-known logit dynamics, which were developed in evolutionary game theory

(Fudenberg and Levine (1998) and Sandholm (2009)).

The adjustment process of (2.18) allows us to define stability of long-run equilibrium h* in

the sense of local stability: the stability of the linearized system of (2.18) at h*. It is well

known in dynamic system theory that the local stability of the equilibrium h* is determined by

examining the eigenvalues of the Jacobian matrix of the adjustment process6:

IhvhJhF −∇=∇ )()()( H (2.19)

where each of )(hJ and )(hv∇ is a K-by-K matrix whose ),( ji entry is ji vP ∂∂ /))(( hv

and ji hv /)( ∂∂ h , respectively.

3. Net agglomeration forces in a racetrack economy

3.1. Racetrack economy and spatial discounting matrix

Consider a “racetrack economy” in which 4 regions }3 ,2 ,1 ,0{ are equidistantly located

on a circumference with radius 1. Let ),( jit denote the distance between two regions i and j.

We define the distance between two regions as that measured by the minimum path length:

),()4/2(),( jimjit ⋅= π

where } ||4 , |.{|min),( jijijim −−−≡ . The set )}3,2,1,0,( ),,({ =jijit of the distances

determines the spatial discounting matrix D whose ),( ji entry, ijd , is given by:

)],( )1(exp[ jitdij ⋅⋅−−≡ τσ (3.1a)

Defining the spatial discount factor (SDF) by

)]4/2( )1(exp[ πτσ ⋅−−≡r (3.1b)

we can represent ijd as ),( jimr . It follows from the definition that the SDF r is a

monotonically decreasing function of the transportation cost (technology) parameter τ, and

hence the feasible range of the SDF (corresponding to ∞+<≤ 0 τ ) is given by ]1 ,0( :

1 0 =⇔= rτ , and 0 →⇔+∞→ rτ . Note here that the SDF yields the following

expression for the SDM in the racetrack economy:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

1

1

1

1

2

2

2

2

rrr

rrr

rrr

rrr

D

6 See, for example, Hirsch and Smale (1974).

10

As easily seen from this expression, the matrix D is a circulant which is constructed from the

vector T2

0 ],,,1[ rrr≡d (see Appendix 3 for the definition and properties of circulant

matrices). This circulant property of the SDM plays a key role in the following analysis.

3.2. Stability, eigenvalues and Jacobi matrices

Stability of equilibrium solutions for the CP model can be determined by examining the

eigenvalues T

321,0 ],,[ gggg≡g of the Jacobian matrix of the adjustment process (2.18).

Specifically, the equilibrium solution h satisfying (2.17) is asymptotically stable if all the

eigenvalues of )( hF∇ have negative real parts; otherwise the solution is unstable (i.e., at

least one eigenvalue of )( hF∇ has a positive real part), and the solution moves in the

direction of the corresponding eigenvector. The eigenvalues, if they are represented as

functions of the key parameters of the CP model (e.g. transport technology parameter τ ),

further enable us to predict whether or not a particular agglomeration pattern (bifurcation) will

occur with changes in the parameter values.

The eigenvalues g of the Jacobian matrix )( hF∇ at an arbitrary distribution h of the

skilled labor cannot be obtained without resorting to numerical techniques. It is, however,

possible in some symmetric distributions h to obtain analytical expressions for the

eigenvalues g of the Jacobian．The key tool for making this possible is a circulant matrix,

which has several useful properties for the eigenvalue analysis. To take “Property 1” of a

circulant in Appendix 3 for example, it implies that if )( hF∇ is a circulant then the

eigenvalues g can be obtained by discrete Fourier transformation (DFT) of the first row vector

0x of )( hF∇ : 0 xZg = , where Z is a 4-by-4 DFT matrix. Furthermore, “Property 2” assures

us that )( hF∇ is indeed a circulant if )(hJ and )( hv∇ in the right-hand side of (3.2) are

circulants (note here that a unit matrix I is obviously a circulant).

A uniform distribution of skilled workers, ]4/ ,4/ ,4/ ,4/[ HHHH≡h , which has

intrinsic significance in examining the emergence of agglomeration, gives us a simple

example for illustrating the use of the above properties of circulants. We first show below that

the Jacobian matrix )( hF∇ at the uniform distribution h is a circulant. This in turn allows

us to obtain analytical expressions for the eigenvalues of )( hF∇ as will be shown in 3.3.

For clarity of exposition, we restrict the analysis below to the case for the Pf model while the

same conclusion holds for the FO model (for the details, see Appendix 2).

In order to show that )( hF∇ is a circulant, we examine each of )(hv∇ and )(hJ in

turn. For the configuration h in which 4/Hh ≡ skilled workers are equally distributed in

each region (i.e., ],,,[ hhhh≡h ), the definition of M in (2.13) yield DhM1)()( −= hd , where

1d ⋅≡ 0d , and hence the Jacobian matrix of indirect utility functions at h reduces to:

}]/[ ]/[ {)( 21 dadbh DDhv −=∇ − (3.2)

where a and b are constant parameters defined as:

11

)1( 11 −− +≡ ha σ , 11)1( −− +−≡ σσb (3.3)

Note that the right-hand side of (3.2) consists only of additions and multiplications of the

circulant matrix D. It follows from this that )(hv∇ is a circulant. We now show that )(hJ

is a circulant. From the definition (2.16) of the location choice probability functions P(h), we

have the Jacobian matrix J(h) at h as:

( )EIhJ )4/1()4/()( −= θ (3.4)

where E is a 4 by 4 matrix whose entries are all equal to 1. This clearly shows that )(hJ is a

circulant because I and E are obviously circulants. Thus, both )(hv∇ and )(hJ are

circulants, and this leads to the conclusion that the Jacobian matrix of the adjustment process

at the configuration h :

IhvhJhF −∇=∇ )()()( H (3.5)

is a circulant.

3.3. Net agglomeration forces

The fact that the matrices )(hJ and )(hv∇ as well as )( hF∇ are all circulants allows

us to obtain the eigenvalues g of )( hF∇ by applying a similarity transformation based on

the DFT matrix Z. Specifically, the similarity transformation of both sides of (3.5) yields:

][diag][diag ][diag][diag 1eg −= δH (3.6a)

where δ and e are the eigenvalues of )(hJ and )(hv∇ , respectively. In a more concise

form this can be written as:

1eg −⋅= ][ ][ δH (3.6b)

where ][][ yx ⋅ denote the component-wise products of vectors x and y. The two eigenvalues,

δ and e , in the right-hand side of (3.6) can easily be obtained as follows. The former

eigenvalues δ are readily given by the DFT of the first row vector of )(hJ in (3.4):

T ]1 ,1 ,1 ,0[)4/(θ=δ (3.7)

As for the latter eigenvalues e , notice that )(hv∇ in (3.2) consists of additions and

multiplications of the circulant D/d. This implies that the eigenvalues e can be represented as

functions of the eigenvalues f of the spatial discounting matrix D/d:

}][][{ 2

1 ffe abh −= − (3.8)

where 2][x denotes the component-wise square of a vector x (i.e., ][][][ 2

xxx ⋅≡ ). The

eigenvalues T

321,0 ],,[ ffff≡f , in turn, are obtained by the DFT of the first row vector

d/0d of the matrix d/D :

12

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

==

)(

)(

)(

1

1

i1i1

1111

i1i1

1111

1/ 220

rc

rc

rc

r

r

r

dddZf (3.9)

where i denotes the imaginary unit, )1/()1()( rrrc +−≡ . Thus, we have the following

proposition characterizing the eigenvalues and eigenvectors of the Jacobian matrix )(hF∇ :

Proposition 2: Consider a uniform distribution ],,,[ hhhh=h of skilled workers in a

racetrack economy with 4 regions. The Jacobian matrix )(hF∇ of the adjustment process

(2.18) of the CP model at h has the following eigenvector and the associated eigenvalues:

1) the kth

eigenvector )3,2,1,0( =k is given by the kth

row vector, kz , of the discrete

Fourier transformation (DFT) matrix Z.

2) the kth

eigenvalue )3,2,1( =kgk is given by a quadratic function of the kth

eigenvalue fk

of the spatial discounting matrix d/D :

))(( rfGg kk θ= )3,2,1( =k , 10 −=g (3.10a)

12 )( −−−≡ θxaxbxG (3.10b)

)1/()1()()()( 31 rrrcrfrf +−≡== , 22 )()( rcrf = (3.10c)

where a and b are constant parameters defined in (3.3) for the Pf model (For the FO model,

see Appendix 2).

The eigenvectors )}3,2,1,0( ,{ =kkz in the first part of Proposition 2 represent

agglomeration patterns of skilled workers by the configuration pattern of the entries. For

example, all entries of z0 are equal to one, and the entry pattern of ]1,1,1,1[ 0 =z corresponds

to the state (configuration of skilled workers among four regions) in which skilled workers are

uniformly distributed among four regions; ]1,1,1,1[ 2 −−=z has the alternate sequence of 1

and −1 representing a duocentric pattern in which skilled workers reside in two regions

alternately; similarly, 1z and 3z correspond to a monocentric pattern.

The eigenvalue gk in the second part of the proposition can be interpreted as the strength of

“net agglomeration force” that leads the uniform distribution in the direction of the kth

agglomeration pattern (i.e., the kth

eigenvector). By the term “net agglomeration force”, we

mean the net effect of the “agglomeration force” minus “dispersion force”. Specifically, each

of these two forces corresponds to xb and 2 xa in (3.10b), respectively. As is clear from

the derivation of the eigenvalues g, the former term ( xb ) stems from S∇ and the first term T

M of )(Hw∇ (see (A1.2) and (A1.3) in Appendix 1), each of which means the so-called

“forward linkage” (or “price-index effect”) and “backward linkage” (or “demand effect”),

respectively. This implies that this term represents the centripetal force induced by the

increase in variety of products that would be realized when the uniform distribution h

13

deviates to the agglomeration pattern zk. The latter term ( 2 xa ), which stems from )(Lw∇

and the second term of )(Hw∇ , represent the centrifugal force due to the increased market

competition (“market crowding effect”) in the agglomerated pattern zk.

4. Theoretical prediction of agglomeration patterns

4.1. Emergence of agglomeration

In order for the bifurcation from the uniform equilibrium distribution ],,,[ hhhh=h to

occur with the changes in the SDF r (or the transportation cost τ ), either of the eigenvalues

)( 31 gg = and 2g must change sign. Since the eigenvalues )3,2,1( =kgk are given by

)( kfG , the changes in sign mean that there should exist real solutions for the quadratic

equation (4.1) with respect to kf :

0)( 12 =−−≡ −θkkk fafbfG (4.1)

Moreover, the solutions must lie in the interval )1 ,0[ that is the possible range of the

eigenvalue fk (see (3.10)). These conditions lead us to the following proposition:

Proposition 3: Suppose that we continuously increase the value of the SDF r (i.e., we

decrease the transport cost parameter τ ) of the CP model in a racetrack economy with 4

regions, starting from 0=r . In order for a bifurcation from a uniform equilibrium

distribution ],,,[ hhhh=h to some agglomeration to occur, the parameters of the CP model

should satisfy:

0 4 12 ≥−≡Θ −θab , and ab 2≤Θ+ (4.2)

The first inequality of (4.2) is the condition for the existence of real solutions of the (4.1)

with respect to fk . This condition is not necessarily satisfied for the cases when heterogeneity

of consumers is very large (i.e., θ is very small), which implies that no agglomeration

occurs in such cases. The second inequality of (4.2), which stems from the requirement that

the solutions of (4.1) are less than 1, corresponds to the “no black-hole” condition that is well

known in literature dealing with the two-region CP model. For the cases when parameters of

the CP model do not satisfy this condition, the eigenvalues )3,2,1( =kgk are positive even

when the SDF is zero (i.e., transportation cost τ is very high), which means that the uniform

distribution h cannot be a stable equilibrium.

In the following analyses, we assume that the parameters ),,( θσh of the CP model satisfy

(4.2). We then have two real solutions for the quadratic equation (4.1) with respect to kf :

)2/()(* abx Θ+≡+ and )2/()(* abx Θ−≡− (4.3)

Each of the solutions *

±x means a critical value (“break point”) at which a bifurcation from

the flat earth equilibrium h occurs when we regard the eigenvalue fk as a bifurcation

parameter. Since we are interested in the process of increasing the SDF r, it is more

14

meaningful to express the break point in terms of the SDF (rather than in terms of fk). Note

here that each of the eigenvalues )}({ rfk is a monotonically decreasing function of the SDF

(see (3.10c) in Proposition 2). This implies that, in the process of increasing the SDF, the

eigenvalue )(rfk first crosses the critical value *

+x before reaching *

−x . We can also see

from (3.10c) that the eigenvalue f2(r) first reaches the critical value *

+x before )(1 rf does,

since )(2 rf is always smaller than )(1 rf :

]1 ,0()()()()( 221 ∈∀≡>≡ rrcrfrcrf

Thus, we can conclude that, in the course of the steady increases in the SDF, the first

bifurcation occurs when the SDF first reaches the critical value *

+r that satisfies:

2**

2* )()( +++ ≡= rcrfx

To be more specific, the critical value *

+r of the SDF is given by:

)1( / )1( ***+++ +−= xxr (4.4)

It is worth noting that (4.4) implicitly provides information on the changes in *

+r with the

changes in values of the CP model parameters ),,( θσh since *

+x is explicitly represented as

a function of the CP model parameters in (4.3). To take but one example, consider the effect

of an increase in the skilled workers h of the Pf model with homogeneous consumers (i.e.,

+∞→θ ), for which the solution *

+x in (4.3) reduces to:

)1(// .lim 1

* −++∞→

+== hbabx σθ

It can easily be seen that the increase in the number of skilled workers h increases *

+x , and

this in turn decreases the critical value *

+r .

The fact that the eigenvalue f2(r) first crosses the critical value *

+x also enables us to

identify the associated agglomeration pattern that emerges at the first bifurcation. Recall here

that the equilibrium solution moves in the direction of the eigenvector whose associated

eigenvalue first hits the critical value. As stated in Proposition 2, this direction is the second

eigenvector ]1,1,1,1[ 2 −−=z . Therefore, the pattern of agglomeration that first emerges is:

] , , ,[2 δδδδδ −+−+=+= hhhhzhh )0( h≤≤ δ

in which skilled workers agglomerate in alternate two regions. Thus, we can characterize the

bifurcation from the uniform distribution as follows.

Proposition 4: Suppose that the conditions of (4.2) in Proposition 3 are satisfied for the CP

model, and that the uniform distribution ] , , ,[ hhhh=h of skilled workers is a stable

equilibrium at some value of the SDF r. Starting from this state, we consider the process

where the value of the SDF continuously increases (i.e., the transportation cost τ decreases).

1) The net agglomeration force (i.e., the eigenvalue) kg for each agglomeration pattern

(i.e., the eigenvector) zk increases as the SDF increases, and the uniform distribution

15

becomes unstable (i.e., agglomeration emerges) at the break point *

+= rr given by (4.3)

and (4.4).

2) The critical value *

+r for the bifurcation decreases, as a) the heterogeneity of skilled

workers (in location choice) is smaller (i.e., θ is large), b) the number of skilled workers

relative to that of the unskilled workers are larger (i.e., h is large), c) the elasticity of

substitution between two varieties is smaller (i.e., σ is small).

3) The pattern of agglomeration that first emerges is ] , , ,[ δδδδ −+−+= hhhhh

)0( h≤≤ δ , in which skilled workers agglomerate alternately in two regions.

4.2. Evolution of agglomeration

In conventional CP models with two regions, increases in the SDF (or decreases in

transportation cost τ) lead to the occurrence of a bifurcation from the uniform distribution h

to a monocentric agglomeration. In the CP model with four (or more) regions, the first

bifurcation shown in 4.1 does not directly branch to the monocentric pattern; instead, further

bifurcations (leading either to a more concentrated pattern or a more dispersed pattern) can

repeatedly occur. In what follows, we will examine such evolution of agglomeration after the

first bifurcation, restricting ourselves to the homogeneous consumer case (i.e., +∞→θ ).

4.2.1. Evolution to a duocentric pattern h*–Sustain point for h

*

For the Pf model with homogeneous consumers, the deviation δ from the uniform

distribution h monotonically increases with the increases in the SDF after the first

bifurcation. Shortly after the increases in the SDF from the break point *

+= rr , this leads to a

duocentric pattern, ]0 ,2 ,0 ,2[* hh=h , where skilled workers equally exist only in the

alternate two regions. The fact that the duocentric pattern h* may exist as an equilibrium

solution of the Pf model can be confirmed by examining the “sustain point” for h*. The

sustain point is the value of the SDF above which the equilibrium condition for h*,

)}({max)()( **2

*0 hhh k

kvvv == (4.5)

is satisfied. As shown in Appendix 4, the condition of (4.5) indeed holds for any r larger than *

01r , which is the sustain point for h*. This is illustrated in Figure 1(a), where the horizontal

axis denotes the SDF r, and the curve represents the utility difference )()( *

1

*

0 hh vv − as a

function of r. As can be seen from this figure, )()( *

1

*

0 hh vv − , is positive for any r larger than

1.0=r (the sustain point), which means that the duocentric pattern h* continues to be an

equilibrium for the range of the SDF above the sustain point.

4.2.2. Bifurcation from the duocentric pattern–Break point at h*

After the emergence of the duocentric pattern ]0 ,2 ,0 ,2[* hh=h , further increases in the SDF

(above the sustain point *

01rr = ) can lead to further bifurcations (i.e., *h become unstable).

In order to investigate such a possibility, we need to obtain the eigenvectors and the

associated eigenvalues for the Jacobian matrix of the adjustment process at *h :

16

‐1.0 

‐0.8 

‐0.6 

‐0.4 

‐0.2 

0.0 

0.2 

0.4 

0.6 

0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0 

V0‐V1

‐1.5 

‐1.0 

‐0.5 

0.0 

0.5 

1.0 

1.5 

0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0 

V0‐V1

V0‐V2

(a) The sustain point for *h (b) The sustain point for **

h

1. Figure The sustainable regions for ]0 ,2 ,0 ,2[* hh=h and ]0 ,0 ,0 ,4[** h=h

IhvhJhF −∇=∇ )( )()( **

* H

A seeming difficulty we encounter in obtaining the eigenvalues is that the Jacobian )( *hF∇

at *h is no longer a circulant matrix, unlike the Jacobian )( hF∇ at h . This is because of

the loss of symmetry in the configuration of skilled workers: D4 symmetry of h is reduced

to D2 symmetry of *h (see Figure 2), which leads to the fact that )( *

hJ and )( *hv∇ are

not circulants. However, as it turns out, it is still possible to find a closed form expression for

the eigenvalues of )( *hF∇ by using the fact that the duocentric pattern *

h has partial

symmetry and the submatrices of )( * hJ and )( *

hv∇ are circulants (see Lemma A.1).

D4 symmetry D2 symmetry

2. Figure Symmetry of the two configurations ) , , ,( hhhh=h and )0 ,2 ,0 ,2(* hh=h

In order to exploit the symmetry remaining in the duocentric pattern *h , we begin by

dividing the set }3,2,1,0{ =C of regions into two subsets: the subset }2,0{ 0 =C of regions

with skilled workers, and the subset }3,1{ 1 =C of regions without skilled workers.

Corresponding to this division of the set of regions, we consider the following permutation

σ of the set }3,2,1,0{ =C :

3)3( ,1)2( ,2)1( ,0)0( ==== σσσσ

so that the first half elements )}1( ),0({ σσ and the second half elements )}3( ),2({ σσ of the

set )}3( ),2( ),1( ),0({ σσσσ=PC correspond to the subsets 10 and CC , respectively. For this

permutation, we then define the associated permutation matrix P:

*01r

**02r

10 vv − ivv −0

r

r

20 vv − 10 vv −

17

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1000

0010

0100

0001

P

It can be readily verified that, for a 4 by 4 matrix A whose ),( ji element is denoted as ija , T

PAP yields a matrix whose ),( ji element is )( )( jia σσ , and that IPPPP == TT ; that is,

the similarity transformation TPAP gives a consistent renumbering of the rows and columns

of A by the permutation σ .

The permutation σ (or the similarity transformation based on the permutation matrix P)

constitutes a “new coordinate system” for analyzing the Jacobian matrix of the adjustment

process. Under the new coordinate system, the SDM D can be represented as:

⎥⎦

⎤⎢⎣

⎡=≡×

)0()1(

)1()0(T

DD

DDPDPD (4.6)

where each of the submatrices D(0)

and D(1)

is a 2-by-2 circulant generated from a vector T2)0(

0 ] ,1 [ r≡d and T

)1(

0 ] ,[ rr≡d , respectively. Similarly, the Jacobian matrix )( *hF∇ of the

adjustment process is transformed into:

IhvhJPhFPhF −∇=∇≡∇ ××× )( )()()( **

T** H (4.7a)

where the Jacobian matrices in the right-hand side are respectively defined by:

⎥⎦

⎤⎢⎣

⎡=∇≡∇×

)11()10(

)01()00(T

*

*

2

1)()(

VV

VVPhvPhv

h (4.7b)

⎥⎦

⎤⎢⎣

⎡=≡×

)11()10(

)01()00(T

*

* )()(

JJ

JJPhJPhJ (4.7c)

and the submatrices )1,0,( and )()( =jiijij

JV are 2-by-2 matrices.

For these Jacobian matrices under the new coordinate system, we can show (Lemma A.1)

that all the submatrices )1,0,( and )()( =jiijij

JV are circulants. This fact allows us to

conclude (Lemma A.2) that knowing only the eigenvalues )00(e of the submatrix )00(

V

)},( /{ 0Cjidhdv ji ∈= is sufficient to obtain the eigenvalues *g of the Jacobian )( *

hF∇ .

Furthermore, it can be readily shown that the submatrix V(00)

is a circulant consisting only of

submatrices D(0)

and D(1)

of the SDM D (these are also circulants), which means that we can

obtain analytical expressions for the eigenvalues )00(e . These considerations lead to the

following lemma:

Lemma 1: The Jacobian matrix )( *hF∇ of the adjustment process (2.18) of the CP model at

h* has the following eigenvector and the associated eigenvalues:

1) the kth

eigenvector )3,2,1,0( =k is given by the kth

row vector, *

kz , of the discrete

Fourier transformation (DFT) matrix PZZPZ ],[diag [2][2]T* ≡ , where ⎥

⎦

⎤⎢⎣

⎡−

≡11

11[2]Z .

18

2) the eigenvalues )}3,2,1,0( { * =kgk are given by )3 ,1 ,0( 1* =−= kgk , and

))(( 2*

*

2 rcGg θ= ,

(4.9a)

12*

* )( −−−≡ θxaxbxG , (4.9b)

where )1/()1()( 222 rrrc +−≡ , and the parameters a* and b are defined as

))2(1( 11* −− +≡ ha σ , 11)1( −− +−≡ σσb .

Proof: see Appendix 5.

The eigenvalues *g obtained in Lemma 1 allow us to determine the critical value (“break

point”) at which a bifurcation from the duocentric pattern ]0 ,2 ,0 ,2[* hh=h to a more

concentrated pattern occurs. In a similar manner to the discussion for the first bifurcation

from the uniform distribution ] , , ,[ hhhh=h , we see that the second bifurcation from the

duocentric pattern h* occurs when the eigenvalue )(*

2 xg changes sign. Since )(*

2 xg is

given by (4.9), the critical values of )( 2rcx ≡ at which the eigenvalue changes sign are the

solutions of the quadratic equation 0)(* =xG . For the homogeneous consumer case (i.e.,

+∞→θ ), the quadratic equation reduces to:

0 )( .lim 2** =−=+∞→

xaxbxGθ

(4.10)

and hence, the critical values (the two solutions of (4.10)) are given by:

))2(1(// 1

*** −+ +== hbabx σ , and 0** =−x (4.11)

Note here that )( 2rcx ≡ is a monotonically decreasing function of the SDF. This implies

that, in the course of increasing the SDF, the x first crosses x+**

. Therefore, the second

bifurcation occurs when the SDF first reaches the critical value **

+r that satisfies )(2 ****

++ = rcx .

That is, the critical value of the second bifurcation in terms of the SDF is given by:

2/1 ****** )]1( / )1[( +++ +−= xxr (4.12)

We can also identify the associated agglomeration pattern that emerges at this bifurcation. The

moving direction away from h* at this bifurcation is the second eigenvector )0 ,1 ,0 ,1(*

2 -=z .

Accordingly, the emerging spatial configuration is given by:

)0 ,2 ,0 ,2(*2

* δδδ −+=+= hhzhh )20( h≤≤ δ

Thus, the properties of the second bifurcation can be summarized as follows:

Proposition 5: Suppose that the SDF r is larger than the sustain point *

01r of the duocentric

pattern ]0 ,2 ,0 ,2[* hh=h and *h is a stable equilibrium for the CP model with

homogeneous consumers. With the increases in the SDF, the duocentric pattern *h become

unstable at the second break point **

+= rr given by (4.11) and (4.12), and then a more

concentrated pattern ]0 ,2 ,0 ,2[ δδ −+= hhh )20( h≤≤ δ emerges.

19

4.2.3. Evolution to a monocentric pattern h**

–Sustain point for h**

After the second bifurcation, the deviation δ from the duocentric pattern

]0 ,2 ,0 ,2[* hh=h monotonically increases with the increase in the SDF, which leads to a

monocentric pattern, ]0 ,0 ,0 ,4[** h=h . This fact can be confirmed by examining a sustain

point for the monocentric pattern. As shown in Appendix 4, the equilibrium condition for **

h :

)}({max)( ****0 hh k

kvv = (4.13)

is satisfied for any r larger than some critical value **

02r , which is the sustain point for **h .

This is illustrated in Figure 1(b), where the horizontal axis denotes the SDF, and each of the

red and blue curves is the utility difference )()( **

2

**

0 hh vv − and )()( **

1

**

0 hh vv − as a

function of r, respectively. As can be seen from this figure, v0(h**

) gives the largest utility

among )}3,2,1,0( )( { ** =ivi h for any r larger than 0.3 (the sustain point), which means that

the monocentric pattern **h continues to be an equilibrium

for the range **

021 rr >> .

The results obtained so far can be summarized as a schematic representation in Figure 3.

.3 Figure A series of agglomeration patterns that emerge in the course of increasing the SDF.

4.3. Collapse of agglomeration

In 4.1, we derived two critical values (of the eigenvalue fk of the matrix D), *

+x and *

−x , at

each of which a bifurcation from the uniform distribution ] , , ,[ hhhh=h occurs, and only the

properties of the bifurcation at the former critical value )( *

+x have been shown. We now

examine the bifurcation at the latter critical value )( *

−x . Before starting a detailed discussion

on the bifurcation at *

−x , we should recall that the eigenvalue gk for each agglomeration

pattern k is a quadratic function of fk as seen in Proposition 2. It follows that each of the

}{ kg is a unimodal function of the SDF because fk is a decreasing function of the SDF. In

other words, all the net agglomeration forces monotonically decrease7 after a monotonic

increasing process in the course of increasing the SDF. This fact implies that agglomeration

7 As seen from (3.12b), the absolute values of both the agglomeration force (the first term, b fk, of gk) and

dispersion force (the second term, a fk2, of gk) decline as fk decreases. It should be noted that the rate of decay

(with the decrease in fk) of the agglomeration force, b, is constant, while that of the dispersion force, 2a fk, is

proportional to fk. This means that the latter gets smaller than the former for fk < b/(2a). In other words, for the

range [rk-1(b/(2a)), 1] of the SDF, the agglomeration force declines faster than the dispersion force (i.e., the net

agglomeration force decreases) as the SDF increases.

r *+r **

+r *01r

**02r 1 0

20

patterns observed in 4.1 and 4.2 might revert to the dispersed distribution h if every net

agglomeration force at h could decrease to negative values for a (relatively) high SDF range.

Whether or not this can happen decisively depends on the value of *

−x , which in turn depends

on the heterogeneity of the consumers. Thus, we divide the following discussion into two

cases: the first is when consumers are homogeneous with respect to location choice, and the

second is when consumers are heterogeneous. As we shall see below, these two cases exhibit

significantly different properties for the bifurcation at *

−x .

For the homogeneous consumers case (i.e., +∞→θ ), 2b=Θ holds and hence the critical

value given by (4.3) reduces to the following simpler expression:

0)( .lim * =−+∞→

θθ

x

Substituting this into the inverse function, )(⋅kr , of the eigenvalue fk(r), we see that

krxr kk ∀==− 1)0()( *

This means that the net agglomeration forces g are always positive for the interval )1 ,[ *

+r of

the SDF. This fact is illustrated in Figure 4(a), where the horizontal axis denotes the SDF r,

and the red curve denotes the eigenvalue g2 as a function of r, while the blue curve denotes

the eigenvalue g1; both of the two curves are unimodal functions whose values reach zero at

1=r . Therefore, the critical value of the SDF at which the bifurcation from agglomeration

equilibrium to the flat earth equilibrium occurs is:

1.lim * =−+∞→

rθ

That is, no matter how large the SDF is, agglomeration never breaks down, except for the

maximum limit at 1=r .

For the case in which consumers are heterogeneous (i.e., θ is finite), we see from (4.3)

that the critical value *

−x is always positive regardless of the values of the CP model

parameters, and hence:

kxrk ∀<− 1)( *

This means that the net agglomeration force gk can be negative on the interval ]1 ),([ *

−xrk of

the SDF. This fact is illustrated in Figure 4(b), where the logit choice parameter θ is set

equal to 20; the horizontal axis, the red and blue curves denote the SDF, the eigenvalues g2

and g1, respectively. Unlike the case of the homogeneous consumers, both of the two

eigenvalues reach zero at 1<r . Accordingly, in the course of increasing the SDF, a

bifurcation from some agglomeration pattern to the uniform distribution (“re-dispersion”)

occurs at the critical value:

1)1(/ )1()( *

**1

* <+−== −−−− xxxrr (4.14)

above which all the net agglomeration forces are negative. That is, the economy with

21

heterogeneous consumers necessarily moves from agglomeration to dispersion when the SDF

increases (the transportation cost τ decreases) to a relatively high range. This is a generalized

result of the “bell-shaped ” bifurcation diagram obtained in Tabuchi and Thisse (2002) and

Murata (2003), in which consumers (skilled workers) in the two-region CP model are

supposed to exhibit idiosyncratic taste differences in residential locations.

-100

-50

0

50

100

150

0.0 0.2 0.4 0.6 0.8 1.0

g(1)

g(2) -2

-1.5

-1

-0.5

0

0.5

1

1.5

0.0 0.2 0.4 0.6 0.8 1.0

g(1)

g(2)

(a) 1000,0.2,1.0 === θσh (b) 20,0.2,1.0 === θσh

.4 Figure Eigenvalues g1 and g2 as functions of the spatial discounting factor.

Some remarks regarding the “re-dispersion” are in order. First, for the “re-dispersion” to

occur in the CP model, it is sufficient that the model parameter satisfies 0* >−x . This

requirement, even if the consumers are homogeneous, can be satisfied by adding an extra

negative (constant) term to the function G in (3.10) determining the critical values *−x .

Therefore, we can deduce that “re-dispersion” occurs whenever some dispersion forces, such

as land rent or congestion externality, that increase with agglomeration (but do not depend on

the SDF r) are introduced into the CP model. This deduction is, of course, applicable to the

conventional two-region model as well as the current four-city model. This gives a consistent

theoretical explanation of several “re-dispersion” results reported in studies on the two-region

CP model: Tabuchi (1998), Helpman (1998)8, Alonso-Villar (2007), in each of which

residential land rent (that is an increasing function of the number of skilled workers in each

city) is introduced into the two-region CP model with homogeneous consumers.

Second, somewhat surprisingly, the “re-dispersion” can be observed not only at the critical

value r-* but also in the course of evolving agglomeration; the re-dispersion at r-

* is the last

among multiple re-dispersions in the course of increasing the SDF. The mechanism by which

8 Strictly speaking, the Helpman’s model and Murata-Thisse’s model do not exhibi “re-dispersion”; in their

models, a uniform distribution is unstable (i.e., an agglomerated pattern is a stable equilibrium) when the SDF is

very low (i.e., the transportation cost is very high). This is because their models do not satisfy the “no

black-hole” condition (note that their models are not endowed with immobile laborers that function as a

dispersion force in the Krugman’s CP model), which causes the degeneration of the first bifurcation from the

uniform distribution. As a result, their model exhibits only a single time of bifurcation from the agglomeration to

dispersion, and the one and only bifurcation corresponds to the “re-dispersion” in the current context.

g2

g1

r

gk

*+r

*−r

g2

g1

r

gk

*+r

22

such repetitions of agglomeration and dispersion may occur can best be explained by the

example in Figure 5. Each of the red and the blue curves in this figure respectively represent

the eigenvalue g2 and g1 at the uniform distribution h as a function of the SDF r (the

horizontal axis). It follows that a bifurcation from agglomeration to dispersion occurs at the

time if all the eigenvalues become negative. We can see from the figure that this actually

occurs twice during the process of increasing r:

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.0 0.2 0.4 0.6 0.8 1.0

g(1)

g(2)

.5 Figure Repetition of agglomeration and dispersion ( 20,4.2,1.0 === θσh )

:A Range the uniform distribution h is unstable because the eigenvalue g2 associated

with the eigenvector ]1,1,1,1[ 2 −−=z at h is positive.

:B Range the uniform distribution h is stable because all the eigenvalues }{ kg at h

are negative. The bifurcation from agglomeration )( 2 zh δ+ to dispersion occurs at the

boundary **

−r between A and B.

:C Range the uniform distribution h is unstable because the eigenvalues g1 and g3

associated with the eigenvectors ]i,1,i ,1[ 1 −−=z and ]i ,1,i,1[ 3 −−=z at h are

positive. The bifurcation from dispersion to agglomeration ))(( 3 1 zzh ++ δ occurs at the

boundary r+**

between B and C.

:D Range the uniform distribution h is stable because all the eigenvalues }{ kg at h

are negative. The bifurcation from agglomeration ))(( 3 1 zzh ++ δ to dispersion occurs at

the boundary *

−r between C and D.

The observations so far can be summarized as the following proposition:

Proposition 6: Starting with an agglomeration state, we consider the process where the SDF

continuously increases (i.e., the transportation cost τ decreases).

1) The net agglomeration force kg for each kz monotonically decreases with the increase

in the SDF after a monotonic increase process (i.e., gk is a unimodal function of the SDF).

2) For the CP model with homogeneous consumers, all the net agglomeration forces reach

zero only at the limit of 1=r . That is, the agglomeration equilibrium never reverts to the

uniform distribution equilibrium during the course of increasing the SDF.

g2

g1

Range A

Range B

Range C

Range D

r

gk

*+r *

−r **

+r **−r

23

3) For the CP model with heterogeneous consumers, all the net agglomeration forces

reaches zero at a strictly positive value of the SDF. That is, a bifurcation from some

agglomeration to the uniform distribution equilibrium (“re-dispersion”) occurs at *

−= rr ,

where *

−r is given by (4.3) and (4.14). Furthermore, the “re-dispersion” can occur not only

from the monocentric agglomeration but also from the duocentric agglomeration. That is,

repetitions of agglomeration and dispersion may be observed for the CP model with

heterogeneous consumers (see Figure 5)

5. Concluding remarks

This paper provided a simple approach to analyzing the bifurcation phenomena in the

multi-regional core-periphery (CP) model. The proposed method allows us not only to

examine whether or not agglomeration of mobile factors emerges from a uniform distribution,

but also to trace the evolution of spatial agglomeration patterns (i.e., bifurcations from various

polycentric patterns as well as a uniform pattern) with the steady decreases in transportation

cost. Furthermore, it is theoretically deduced that the evolutionary process in the CP model

may exhibit repetitions of agglomeration and dispersion, which gives a theoretical explanation

for several “bell-shaped development” results reported in studies on the two-region CP

models. Although we restricted ourselves to illustrating the evolutionary process in the

four-region case, the approach can be extended to more general cases. The detailed discussion

on the method available for models with an arbitrary number of regions can be found in

Akamatsu et al., (2009).

Through the analysis of the CP model, we also demonstrated that the spatial discounting

matrix (SDM) encapsulates the essential information required for analyzing the multi-regional

CP model. Specifically, in order to know spatial concentration-dispersion patterns that may

emerge in the CP model, it suffices a) to obtain the eigenvalues of the SDM, and b) to

represent the Jacobian matrix of the indirect utility as a function of the SDM. It should be

emphasized that this fact holds for a wide variety of models dealing with the formation of

spatial patterns in economic activities. Indeed, the same procedure as that of this paper can be

readily applied to other variants of the multi-regional CP model. Therefore, it would be

valuable for future research to investigate the bifurcation behaviors in these models by

exploiting the method presented in this paper. We believe that such systematic studies would

substantially improve our understanding of the nature of agglomeration economies.

Acknowledgements

We are grateful for comments and suggestions made by Masahisa Fujita, Kiyohiro Ikeda,

Tomoya Mori, Se-il Mun, Yasusada Murata, Toshimori Otazawa, Daisuke Oyama, Yasuhiro

Sato, Takatoshi Tabuchi, Kazuhiro Yamamoto, Dao-Zhi Zeng, as well as participants at the

24

23th Applied Regional Science Conference Meeting in Yamagata. Funding from the Japan

Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B)

19360227/21360194 is greatly acknowledged.

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Appendix 1 Jacobian Matrices for the Pf model

The Jacobian matrix of the adjustment process for the Pf model is

IhvhJhF −∇=∇ )()()( H .

The Jacobian matrices, vJ ∇ and , in the right-hand side of (2.19) is given by

ji

ji

PP

PP

V

P

V

P

ji

ji

j

i

j

i

=≠

⎩⎨⎧

−−

=∂∂

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

≡if

if

)1(,

)()(

θθV

hJ . (A1.1)

)]()([)()( )()(1hwhwhShv

HL ∇+∇+∇=∇ −σ (A1.2)

where the matrices )( , LwS ∇∇ and )(Hw∇ in the right-hand of (A1.2) are given by

T1)1( MS−−=∇ σ , (A1.3a)

T)( MMw −=∇ L , T)( MMHMw −=∇ H . (A1.3b)

Appendix 2 Jacobian Matrices and Its Eigenvalues for the FO model

We show the Jacobian matrix )(hv∇ and eigenvalues g of )(hF∇ for the FO model in turn.

(1) Differentiating the indirect utility function (2.14) with respect to h and substituting hh = into the

result, )(hv∇ is given by:

)( )/()( 11hwDhv ∇+=∇ −

−− wdh κ (A2.1)

where )1/( −≡− σμκ and ])1/[()( hww i κκ −=≡ h . The matrix )(hw∇ in the right-hand side is

obtained by differentiating both sides of (2.9) with respect to h and substituting hh = into this equation:

])/([ )/(])/( [ )( 11 dddwh DIDDIhw −−=∇ −− κκ (A2.2)

where σμκ /≡ . Substituting (A2.2) into (A2.1), we obtain the Jacobian matrix of the indirect utility:

} )]/()[/()]/([)/({ )( 1

1 ddddh DIDDIDhv −−+=∇ −−

− κκκ (A2.3)

(2) To obtain g, it is sufficient to calculate the eigenvalues e of )(hv∇ , since )(hF∇ and )(hJ are

expressed as (3.5) and (3.4), respectively. From the fact that the right-hand side of (A2.3) consists only of D

and I (i.e., )(hv∇ is a circulant), we can obtain e by the DFT of the first row vector of this matrix.

]}[]1[{ 11

kkkkk ffffhe −−+= −−

− κκκ (A2.4)

Substituting (A2.4) and (3.7) into (3.6b), we have:

0

0

if

if

]1[)(

1

1 ≠

=

⎩⎨⎧

−

−=

− k

k

ffGg

kk

k κθ (A2.5)

where 12 )( −−−≡ θkkk fafbfG , 1+≡ −κκa , )1( 1−

− ++≡ θκκb .

Appendix 3 Properties of Circulant Matrices

A circulant C is defined as a square matrix of the form:

26

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

≡

−−

−

−−

−−

01221

10132

22101

12210

ccccc

ccccc

ccccc

ccccc

KK

K

KK

KK

L

L

MMM

L

L

C

The elements of each row of C are identical to those of the previous row, but are moved one position to the

right and wrapped around. The whole circulant is evidently determined by the first row vector

c=[c0,c1,…,cK-1]. Circulant matrices satisfy the following two well-known properties9.

Property 1: Every circulant matrix C is diagonalized by the following similarity transformation:

)(diag* λCZZ =

where Z is the DFT matrix whose ),( kj entry is given by )]/2(iexp[ Kjkjk πω = , 1i −≡ ;

[ ]T110 ,,, −≡ Kλλλ Lλ , and Z

* denotes the conjugate transpose of Z. The k

th eigenvalues and the

eigenvectors of C are therefore λk and the kth

row of the DFT matrix Z, respectively. Furthermore, λ is

directly given by the DFT of the first row vector c of C: TZcλ = .

Property 2: If C1 and C2 are circulant matrices, the sum C1 + C2 and the product C1C2 are circulants. Also,

if C1 is nonsingular, its inverse C1−1

is a circulant.

Appendix 4 Sustain points for ]0 ,2 ,0 ,2[* hh=h and ]0 ,0 ,0 ,4[** h=h

We will show the derivation of sustain points for ]0 ,2 ,0 ,2[* hh=h and ]0 ,0 ,0 ,4[** h=h , in turn.

(1) For the duocentric pattern ]0 ,2 ,0 ,2[* hh=h , we can easily obtain the indirect utility for each region by

substituting *hh = into (2.15):

⎪⎩

⎪⎨⎧

−+++

+−++=

−−−−

−−−

)2ln()1(] ))( )21()(()(2 [

)1ln()1(][1 )(

1111

2111*

rrxhrxh

rhvi

σσ

σσh

)3,1(

)2,0(

==

i

i

where )1/(2)( 2rrrx +≡ . To obtain the sustain point for *h , we represent the utility difference between

the “core” regions and the “periphery” regions as a function of the SDF:

)()()( *1

*001 hh vvrv −≡ )ln()1(2 )21(1)1(2 1

2 xxhxhxh −−−+−−+= σσ

By inspecting the function v01(r), we see that it takes zero value at 1=r

and *

01rr = >0 (i.e., the equation

0)(01 =rv has two positive solutions 1 and *01r ) and that:

⎪⎩

⎪⎨⎧

<≤≥

<<<

1for 0

0for 0)(

*01

*01

01rr

rrrv

This means that the equilibrium condition for *h , )}({max)()( **

2*

0 hhh kk vvv == , is satisfied for any r

larger than *

01r ; that is, *

01rr = is the sustain point for *h .

(2) The sustain point for the monocentric pattern ]0 ,0 ,0 ,4[** h=h can be obtained in a similar manner.

The indirect utility at **h is given by:

⎪⎪⎩

⎪⎪⎨

⎧

−++++

−+++

+

=−−−−

−−−−

−−

rrrrh

rrrrh

h

vi

ln)1(2] )2()(4 [

ln)1(] )()(2 [

]1 [

)(

122211

1111

11

**

σσ

σσ

σ

h

)2(

)3,1(

)0(

===

i

i

i

Define the following utility difference functions at )0 ,0 ,0 ,4(** h=h :

9 For the proofs of these properties, see, for example, Gray (2006).

27

)()()( **1

**001 hh vvrv −≡ )ln()1(2)21(1)1(2 1

2 rrhrhrh −−−+−−+= σσ

)()()( **2

**002 hh vvrv −≡ )ln()1(4)41(1)21(2 221

42 rrhrhrh −−−+−−+= σσ

After a tedious calculation, we can show that:

⎪⎩

⎪⎨⎧

<≤≥

<<<

1for 0

0for 0)(

** 0

** 0

0rr

rrrv

i

i

i )2,1( =i , and **

02**

010 rr <<

where *

0 ir is the solution of 0)( 0 =rv i )2,1( =i . That is, the equilibrium condition for **h , )( **

0 hv

)}({max **hkk v= , is satisfied for any r larger than **

02r ; that is, **

02rr = is the sustain point for **h .

Appendix 5 Proof of Lemma 1

The following two lemmas help us prove Lemma 1:

Lemma A.1: All the submatrices )1,0,( )()( =jiand ijij

JV defined in (4.7) are circulants.

Lemma A.2: The eigenvalues )3,2,1,0( * =kg k of the Jacobian matrix )( *

hF∇ at ]0 ,2 ,0 ,2[* hh=h

are represented as 1 )00(

1 *

2 −= eg θ and )3 ,1 ,0( 1* =−= kg k , where T)00(

1

)00(

0

)00( ] ,[ ee≡e denotes the

eigenvalues of the Jacobian matrix V(00)

.

Proof of Lemma A.1: We prove that each of the submatrices J(ij)

and V(ij)

is a circulant in turn.

(1) Let each of p(0) and p(1) denote the location choice probability of the subset of regions )0(0 =iC and

)1(1 =iC at the agglomerating pattern h*, respectively:

⎩⎨⎧

==−

=−+−

−≡

)1( 2/

)0( 2/)1(

)]}(exp[)]({exp[ 2

)](exp[*

1 *

0

*

)(i

i

vv

vp i

i εε

θθθ

hh

h (A5.1)

A straightforward calculation of the definition (4.7) of T

*

* )()( PhJPhJ ≡× yields:

)()(

)( jii

ji p pJ θ= where ⎥⎦

⎤⎢⎣

⎡−−

−−=

)()(

)()()(

jijj

jjijji

pp

pp

δδ

p )1,0,( =ji

This explicitly shows that the submatrices )1,0,()( =jiijJ are circulants generated from:

⎪⎩

⎪⎨⎧

≠−−

=−−=

)( ],[

)( ],1[

)()()(

)()()( )(

0

jippp

jippp

jji

jjiji

θ

θJ (A5.2)

(2) As is shown in (A1.2), the Jacobian matrix )( *hv∇ consists of additions and multiplications of

1** )}({)( −≡ hΔDhM . It follows from this that the Jacobian matrix T

*

* )()( PhvPhv ∇≡∇× in the new

coordinate system consists of those of T

* )( PhMP , which in turn is composed of submatrices )(ij

M

)1,0,( =ji :

⎥⎦

⎤⎢⎣

⎡≡ −

)11()10(

)01()00(1T

*

)2()(MM

MMPhMP h

Therefore, in order to prove that V(ij)

)1,0,( =ji are circulants, it suffices to show that the submatrices )(ij

M )1,0,( =ji are circulants. Note here that T

* )( PhMP can be represented as:

T

* )( PhMP ])}({][[ T1*

T

PhΔPPDP−= (A5.3)

The first bracket of the right-hand side of (A5.3) is given in (4.6), and a simple calculation of the second

bracket yields:

1)1( )1()0( )0(

1T1* ]},,,[diag{)2()}({ −−− = ddddhPhΔP

28

where 2

)0( 1 rd +≡ and rd 2)1( ≡ . Thus, we have:

⎥⎥⎦

⎤

⎢⎢⎣

⎡= −−

−−−

)0(1

)1()1(1

)1(

)1(1 )0()0(

1 )0(1T

*

)2()(

DD

DDPhMP

dd

ddh (A5.4)

which shows that the submatrices )(ijM )1,0,( =ji are circulants.

Proof of Lemma A.2: We prove that the eigenvalues of the Jacobian )( *hF∇ are given by

1−T)00(1 ]0, ,0,0[ eθ when 2/1)0( →p . Consider the Jacobian )( *

hF×∇ in the new coordinate system:

T

*

* )()( PhFPhF ∇≡∇×IhvhJ −∇= ×× )()( *

*

H IFF

FF−⎥

⎦

⎤⎢⎣

⎡≡

)11()10(

)01()00(

2 (A5.5a)

∑ =≡1,0

)()()(

k

jkkiijVJF )1,0,( =ji (A5.5b)

Note here that the submatrices )(ijF of the Jacobian matrix )( *

hF×∇ are circulants because all

submatrices )(ijJ and )(ij

V )1,0,( =ji are circulants. This enables us to diagonalize each of the

submatrices )(ijF by using a 2-by-2 DFT matrix Z[2]:

Iff

ffZFZ −⎥

⎦

⎤⎢⎣

⎡=∇ ××−×

)(diag)(diag

)(diag)(diag 2 )(

)11()10(

)01()00(1 (A5.6)

where )(ijf is the eigenvalues of )(ij

F , and ],[diag [2][2] ZZZ ≡× . It also follows that applying the

similarity transformation based on ]2[Z to both sides of (A5.5b) yields:

∑ = ⋅=1,0

)()()( ][][k

jkkiijef δ )1,0,( =ji (A5.7)

where T)(

1

)(

0

)( ] ,[ ijijij δδ≡δ and T)(

1

)(

0

)( ] ,[ ijijij ee≡e denote the eigenvalues of the Jacobian matrices )(ijJ

and )(ijV )1,0,( =ji , respectively. The former eigenvalues δ(ij)

can be given analytically by the DFT of the

first row vector )(0ij

J of the submatrix )(ijJ :

)(0]2[

)( ijij JZ=δ )1,0,( =ji (A5.8)

Substituting (A5.1) and (A5.2) into (A5.8), we have:

T)00( ] 1 )[1)(2/( εεθ −=δ ,

T

)11( ] 11 [)2/( εεθ −=δ

T

)01( ] 0 )[1()2/( εεθ −−=δ ,

T

)10( ] 01 [)2/( −= εεθδ

It follows from this that:

⎩⎨⎧ ==

==→→ otherwise

0 if ] 10 )[2/( .lim .lim

T)(

2/1

)(

2/1)0( 0

jiijij

p

θε

δδ

Substituting these δ(ij) )1,0,( =ji into (A5.7) yields:

⎩⎨⎧

=

==

→ 1 if

0 if]0 )[2/( .lim

T)0(1)(

2/1)0( i

ie jij

p 0f

θ )1,0( =j

Thus, when 2/1)0( →p , (A5.6) reduces to:

I

00

ZFZ −

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=∇ ××−× )01(1

)00(1

1 0

00

0

00

)( eeθ

Converting this into the original coordinate system, we obtain:

29

*

1*1T )( ] )[( ZFZPZFZP ∇=∇ −××−×I

0

00

−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

00

)01(1

)00(1 ee

θ

where PZPZ×≡

T* . Since eigenvalues of an upper-triangular matrix are given by the diagonal entries, we

can conclude that the eigenvalues of the Jacobian )( *hF∇ are given by 1−T)00(

1 ]0, ,0,0[ eθ .

Proof of Lemma 1: Substituting (A1.2) and (A5.4) into the definition of )( *hv

×∇ in (4.7), we see that the

Jacobian matrix V(00)

consists of additions and multiplications of D(0)

and D(1)

:

}]/[]/[ {]/[ 2 )1(

)1(

*)1(

2 )0(

)0(*)0(

)0()00( dadadb DDDV +−≡

where 2

)0( 1 rd +≡ , rd 2)1( ≡ , ))2(1( 11* −− +≡ ha σ , 11*)1( )2( −−≡ ha σ , and 11)1( −− +−≡ σσb . Since D

(0)

and D(1)

are circulants, we have the following expressions for the eigenvalues e(00)

of the Jacobian V(00)

:

})()({ 2)1(

*)1(

2)0(

*)0(

)00( fffe aab +−=

⎥⎥⎦

⎤

⎢⎢⎣

⎡≡

)00(1

)00(0

e

e (A5.9)

where each vector )1,0()( =iif is the eigenvalues of D(i)

/d(i), each of which is obtained by DFT of vectors

)(0i

d :

)0(0]2[

)0(

)0(

1dZf

d= ⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−+

=22

1

11

11

1

1

rr⎥⎦

⎤⎢⎣

⎡=

)(

12rc

(A5.10a)

)1(0]2[

)1(

)1(

1dZf

d= ⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡−

=r

r

r 11

11

2

1 ⎥

⎦

⎤⎢⎣

⎡=

0

1 (A5.10b)

Substituting (A5.10) into (A5.9) yields:

0)00(

0 =e , 2*)00(1 xaxbe −= , where )( 2rcx ≡ (A5.11)

Combining (A5.11) and Lemma A.2, we obtain Lemma 1.